Nonnegative Decomposition of Multivariate Information

TL;DR

This paper introduces a new framework for multivariate information decomposition that avoids negative values and clarifies the structure of information shared among sources using a redundancy lattice.

Contribution

It proposes a novel partial information decomposition based on a new redundancy measure, providing a clearer and non-negative interpretation of multivariate information.

Findings

Redundancy lattice clarifies multivariate information structure

Partial information atoms are always non-negative

Interaction information's negativity is due to confounding redundancy and synergy

Abstract

Of the various attempts to generalize information theory to multiple variables, the most widely utilized, interaction information, suffers from the problem that it is sometimes negative. Here we reconsider from first principles the general structure of the information that a set of sources provides about a given variable. We begin with a new definition of redundancy as the minimum information that any source provides about each possible outcome of the variable, averaged over all possible outcomes. We then show how this measure of redundancy induces a lattice over sets of sources that clarifies the general structure of multivariate information. Finally, we use this redundancy lattice to propose a definition of partial information atoms that exhaustively decompose the Shannon information in a multivariate system in terms of the redundancy between synergies of subsets of the sources. Unlike interaction information, the atoms of our partial information decomposition are never negative and always support a clear interpretation as informational quantities. Our analysis also demonstrates how the negativity of interaction information can be explained by its confounding of redundancy and synergy.